There are probably some glaring holes in this list, so feel free to add suggestions! One thing to note, I don't listen to metal so all the Icelandic metal bands (and there are a lot) aren't included.
  1. The Sugarcubes: Life is Good
    2c71f922 1663 44f6 92e4 71b8329a105a
    You really can't talk about contemporary Icelandic music w/o talking about the Sugarcubes. They put Iceland on the map in terms of 80s alt rock. Video: Birthday http://bit.ly/1HoRVg0
  2. Bjork: Homogenic
    9aa9de63 4a8f 4c07 b0cd 9f3d611ff755
    The Sugarcubes begat Bjork. One of the most influential contemporary musicians working right now. Her most recent album Vulnerica is heartbreakingly beautiful. But I still hold Homogenic as her best, blending experimental electronic with pop structured songwriting. Also, it has the song All of Full of Love which is one of my all time faves. Ever. Video: All is Full of Love http://bit.ly/1QUaTNn
  3. Sigur Rós: Með suð í eyrum við spilum endalaust
    647fcc2d e669 42b6 9fc2 0229cb22a43b
    Glorious ethereal post-rock band. Probably the most known group after Bjork and The Sugarcubes. Some of the songs are sung in a made up language ala Cocteau Twins but whatever. It's not like I understand Icelandic anyway. This album is a move away from their minimal mellow work and is rather upbeat. One of the few concerts I've actually been to with my partner @AJ! Video: Inní mér syngur vitleysingur http://bit.ly/1bYqiML
  4. Of Monster and Men: My Head is an Animal
    Eb18f0e2 1705 4c21 b542 3623057c0288
    Their debut album hit the top 10 here in the U.S. which is great. It's a blend of hipster Arcade Fire-esque indie-folk/pop music. Their next album is released this summer. Video: Lakehouse http://bit.ly/1Bbr4wA
  5. Emiliana Torrini: Love in the Time of Science
    0fff4ce2 79f4 4d69 86c5 c3132da9417d
    Her later work gets more folk/folktronica (I hate that term as much as I love the music) but I'm more partial to her earlier trip hop influenced stuff. But all of it is solid. She did the vocals for a Lord of the Rings song and she wrote Kylie Minogue's song Slow. By the way, if you like Emiliana Torrini, check out Firefox AK who is Swedish and has a similar sensibility (though more new wave synth pop influenced). Video: To Be Free http://bit.ly/1PrJPrt
  6. Múm: Finally We are No One
    C2c07d6e e3fd 4ce7 aec4 48a0168bf93f
    Not everything they release works for me. Sometimes it veers into super experimental electronic and I'm probably too dumb to appreciate it. But I love this album. It still borders in glitchy experimental but there are enough sweeping orchestrated songs that grounds it for me. Video: Green Grass of Tunnel http://bit.ly/1A80FoA
  7. Seabear: The Ghost that Carried Us Away
    Bc6ccf3f 90d7 4bc7 926a 32b9b7296f7c
    They're known for a couple of songs being featured on Grey's Anatomy as well as a few other shows. A lot of comparison to Arcade Fire or Sufjan Stevens. Which isn't too far off but also shortchanges them. Video: I Sing I Swim http://bit.ly/1KiLJah
  8. Ólafur Arnalds: For Now I am Winter
    Be95ffd9 6c6c 4986 8a16 f66f1f9813b6
    You might have already heard his work since he gets used in movies and tv shows. If you've seen the trailer for the movie Looper one of his songs is used in it. I'm partially to this album because it's the first time he used a vocalist, his previous work being mostly instrumental electronic compositions. Very soundscape-y soundtrack-y. Video: For Now I am Winter http://bit.ly/1A8cWsY
  9. Retro Stefson: Retro Stefson
    F845dab8 8897 428b b939 ccbdf67deeeb
    They've basically been called the Icelandic Vampire Weekend but I think that kind of lazy. Granted there is the afropop influence in their music but it's less New England college twee and more British neo soul. Video: Glow http://bit.ly/1LbY79S
  10. Ásgeir: In the Silence
    7a35eb3e e088 4ece 8d9f 6eb285e273c7
    Àsgeir is kind of in that Nick Drake/Bon Iver/Jose Gonzalez/Jens Lekman mellow guitar with a guy singer songwriter kind of mold. He got a lit bit of buzz last year but then really came of it. He's only got one album (though it was released twice. Once in Icelandic and once in English for the international markets). It's a solid debut but I think he has potential for something better. Video: King and Cross http://bit.ly/1e9c8dq
  11. Sóley: Ask the Deep
    Eaed76eb fb8b 4692 8dd4 9356ceea9cca
    A former member of Seabear, her second album Ask the Deep was just released a few weeks ago. Haunting songwriting. If you like Lykke Li's more recent (moodier less pop) stuff. Video (audio only): Follow Me Down http://bit.ly/1AkjYuJ
  12. FM Belfast: Don't Want to Sleep
    5b5c8866 1ece 4395 836a 83cbc1287f35
    They have a particular synth-y electro 80s pop feel that a lot of people disparaged but I found kind of chipper and fun. Less experimental than other Icelandic bands but I think they haven't quite hit their stride. Still worth checking out if you dig La Roux, Metric or O Land. Video: Brighter Days http://bit.ly/1L9QC2r
  13. Jónsi: Go
    C13370bf 8723 472a 83b6 198b4613fd30
    As much as I love Sigur Rós, they're so moody! Not this album, by the solo singer Jònsi. Bonus, he sings some of the songs in English! Video: Go Do http://bit.ly/1A8dDTc
  14. Sin Fang: Flowers
    79f8d0c6 b939 464d 8d82 2b472297435b
    Solo project by the guy from Seabear. Decidedly more upbeat than Seabear. If you like that sort of glitch-y plinkerpop music that The German label Morr music is known for (Notwist, Lali Puna, Electric President) this is probably your jam. Of course, if know those bands already, you probably know Sin Fang too. Video: Young Boys http://bit.ly/1GhzbOb
  15. valgeir sigurðsson
    Suggested by @mollyyeh